Other mistake: It is previously explained that Jake cannot grow a moustache, then a few seasons later Jake goes to prison and grows a moustache.
Revealing mistake: In the opening credits (the very beginning), Jake is walking and the image behind him is blurred. When it unblurs (right before it the shot changes), the phone number on the building behind him is 213 which is LA, vs. 718 which is Brooklyn.
Answer: In reality, the Commissioner is a civilian administrator, appointed by the mayor of New York City. He (as of this writing, the role has always been filled by a man; the show gets that part right) is usually someone who has risen through the ranks of the NYPD, but upon becoming Commissioner, is no longer a sworn member of the force (the highest sworn rank is that of Chief). Since Commissioners are appointed, technically anyone can serve in the position, including Deputy Commissioners (who are appointed, in turn, by the Commissioner himself).
Update: On 1 January 2022, Keechant Sewell became the first ever female commissioner of the NYPD.